A typical example of polynomials that have singular points is $y^2 - x^2 - x^3 = 0$, where point $(0,0)$ is the singular point. However, the situation is changed if a linear term is added to this polynomial. Say $y^2 - x^2 - x^3 - x = 0\ $ or $\ y^2 - x^2 - x^3 - t = 0$.
It seems like given a polynomial $f$, the (affine or projective) variety of it, i.e., $V(f)$ won't be singular if $f$ has at least one linear term. I can see why $\ f(x,y,t) = y^2 - x^2 - x^3 - t$ is non-singular, because the partial derivative of $\ f$ with respect to $\ t$ can never be $0$.
But I am wondering how to generalize this to all polynomials in $\ K[x_1, x_2, \cdots, x_n]$? Or is there any material that discusses the patterns on polynomials such that the varieties of them will/won't have singular points?
I know that the Jacobian criteria can do this by checking one point after another. But there could be infinitely many of points to be checked. So I am thinking if there is any such pattern on the polynomials that we can determine the singular points at a high level, or even just for special classes of polynomials.
Moreover, what kind of patterns/characteristics of the polynomial can help determine the dimension of set of singular points?
Please someone guide me to some sources related to this. Thank you very much.